C*-like modules and matrix p-operator norms

We present a generalization of Hölder duality to algebra-valued pairings via Lᵖ L p -modules. Hölder duality states that if p ∈ (1, ∞ ) p ∈ ( 1 , ∞ ) and $$p'$$ p ′ are conjugate exponents, then the dual space of Lᵖ(μ ) L p ( μ ) is isometrically isomorphic to Lp'(μ ) L p ′ ( μ ) . In this work, we study certain pairs (Y,X) ( Y , X ) , as generalizations of the pair (Lp'(μ ), Lᵖ(μ )) ( L p ′ ( μ ) , L p ( μ ) ) , that have an Lᵖ L p -operator algebra-valued pairing Y× X→ A Y × X → A . When the A -valued version of Hölder duality still holds, we say that (Y, X) ( Y , X ) is C*-like. We show that finite and countable direct sums of the C*-like module ( A , A ) are still C*-like when A is any block diagonal subalgebra of d × d d × d matrices. We provide counterexamples when A ⊂ Mdᵖ(C) A ⊂ M d